Symmetric Monoidal Approach To Physical Theories

Symmetric monoidal approach to physical theories

This research area is concerned with providing a logical foundation for physical theories. Recent work in this area has lead to the development of a categorical semantics of quantum mechanics which has particular applications when reasoning about quantum information theory. At the heart of this work lies several full and faithful representations of quantum theory in terms of a picture calculus. This approach was initiated in:

• Samson Abramsky and Bob Coecke, A categorical semantics of quantum protocols, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004).

Some related papers are:

• Samson Abramsky and Ross Duncan: A Categorical Quantum Logic (arXiv:quant-ph/0512114)
• Bob Coecke, Eric Paquette and Dusko Pavlovic: Quantum measurements without sums and POVMs and Naimark's theorem without sums (arXiv:quant-ph/0608035) (arXiv:quant-ph/0608072)
• Peter Selinger: Dagger compact closed categories and completely positive maps (link)
• Jamie Vicary: A categorical framework for the quantum harmonic oscillator (arXiv:0706.0711v2)

Friendly introductions to this approach are found in:

• Bob Coecke, Kindergarten quantum mechanics
• Bob Coecke, Introducing categories to the practicing physicist

More related material is available from Samson Abramsky, Bob Coecke and Peter Selinger's respective homepages. Similar ideas can also be found in work by John Baez, Louis Kauffman and Robin Houston (blog).

• http://arxiv.org/abs/quant-ph/0407224
• http://arxiv.org/abs/quant-ph/0404040
• http://arxiv.org/abs/math.CT/0604542